Equations of Motion and the Hamiltonian Formalism

1992 ◽  
pp. 9-62
Author(s):  
Vladimir E. Zakharov ◽  
Victor S. L’vov ◽  
Gregory Falkovich
Keyword(s):  
Equations Of Motion ◽  
Hamiltonian Formalism

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

scholarly journals Lagrangian and Hamiltonian formalism for constrained variational problems

2002 ◽  
Vol 132 (6) ◽  
pp. 1417-1437 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Critical Points ◽  
Vector Bundles ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Variational Problems ◽  
Linear Constraints ◽  
Legendre Transform ◽  
General Notion ◽  
Horizontal Curves ◽  
Vakonomic Mechanics

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

Modeling of Systems With Position-Dependent Mass Revisited: A Port-Hamiltonian Approach

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Dimitri Jeltsema ◽  
Arnau Dòria-Cerezo
Keyword(s):  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Hamiltonian Approach ◽  
Lagrangian And Hamiltonian Approach ◽  
Modeling Framework ◽  
Straightforward Application ◽  
Composite Energy ◽  
Dependent Mass ◽  
Position Dependent Mass ◽  
Classical Lagrangian

It is known that straightforward application of the classical Lagrangian and Hamiltonian formalism to systems with mass varying explicitly with position may lead to discrepancies in the formulation of the equations of motion. Systems with mass varying explicitly with position often arise from situations where the partitioning of a closed system of constant mass leads to open subsystems that exchange mass among themselves. One possible solution is to introduce additional nonconservative generalized forces that account for these effects. However, it remains unclear how to systematically interconnect the Lagrangian or Hamiltonian subsystems. In this note, systems with mass varying explicitly with position and their properties are studied in the port-Hamiltonian modeling framework. The port-Hamiltonian formalism combines the classical Lagrangian and Hamiltonian approach with network modeling and is applicable to various engineering domains. One of the strong aspects of the port-Hamiltonian formalism is that power-preserving interconnections between port-Hamiltonian subsystems results in another port-Hamiltonian system with composite energy and interconnection structure. The motion of a heavy cable being deployed from a reel by the action of gravity is used as an example.

Introduction

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Mirror Symmetry ◽  
Complex Geometry ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Theoretical Physics ◽  
Lagrange Equations ◽  
Low Dimensional Topology ◽  
The Subject ◽  
Low Dimensional

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

scholarly journals NON-ABELIAN PROCA MODEL BASED ON THE IMPROVED BFT FORMALISM

1998 ◽  
Vol 13 (13) ◽  
pp. 2179-2199 ◽  
Author(s):  
MU-IN PARK ◽  
YOUNG-JAI PARK
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Difficult Problem ◽  
Lagrangian Formulation ◽  
Exponential Form ◽  
Extended Phase ◽  
Physical Variables ◽  
Correction Terms ◽  
Insight Into

We present the newly improved Batalin–Fradkin–Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide a much more simple and transparent insight into the usual BFT method, with application to the non-Abelian Proca model, which has been a difficult problem in the usual BFT method. The infinite terms of the effectively first class contraints can be made to be the regular power series forms by an ingenious choice of Xαβ and ωαβ matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms, is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably, all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space have the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized Stückelberg Lagrangian which is a nontrivial conjecture in our infinitely many terms involved in the Hamiltonian and the Lagrangian.

scholarly journals Transformation properties of the Lagrange function

2008 ◽  
Vol 30 (3) ◽  
pp. 3306.1-3306.8
Author(s):  
Carlo Ferrario ◽  
Arianna Passerini
Keyword(s):  
Equations Of Motion ◽  
Lagrange Function ◽  
Hamiltonian Formalism ◽  
Undergraduate Teaching ◽  
Hamiltonian Mechanics ◽  
Coordinate Transformations ◽  
Lagrangian And Hamiltonian Mechanics ◽  
The Relationship ◽  
Theoretical Mechanics

In the context of the Lagrangian and Hamiltonian mechanics, a generalized theory of coordinate transformations is analyzed. On the basis of such theory, a misconception concerning the superiority of the Hamiltonian formalism with respect to the Lagrangian one is criticized. The consequent discussion introduces the relationship between the classical Hamilton action and the covariance properties of equations of motion, at the level of undergraduate teaching courses in theoretical mechanics.

scholarly journals Spinning bodies in general relativity

2016 ◽  
Vol 13 (08) ◽  
pp. 1640002 ◽  
Author(s):  
J. W. van Holten
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Momentum Tensor ◽  
Curved Space ◽  
Constants Of Motion ◽  
Spinning Bodies ◽  
Compact Spinning ◽  
Independent Derivation

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

scholarly journals Tidal evolution of a close-in planet with a more massive outer companion

2010 ◽  
Vol 6 (S276) ◽  
pp. 508-510
Author(s):  
Adrian Rodríguez ◽  
Sylvio Ferraz-Mello ◽  
Tatiana A. Michtchenko ◽  
Cristian Beaugé ◽  
Octavio Miloni
Keyword(s):  
Numerical Simulation ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Stationary Solutions ◽  
Real System ◽  
Outer Planet ◽  
Combined Effects ◽  
Tidal Evolution ◽  
Tidal Dissipation ◽  
The Mean

AbstractWe investigate the motion of a two-planet coplanar system under the combined effects of mutual interaction and tidal dissipation. The secular behavior of the system is analyzed using two different approaches, restricting to the case of a more massive outer planet. First, we solve the exact equations of motion through the numerical simulation of the system evolution. We also compute the stationary solutions of the mean equations of motion based on a Hamiltonian formalism. An application to the real system CoRoT-7 is investigated.

ON THE FIRST ORDER COVARIANT HAMILTONIAN FORMALISM ON GROUP MANIFOLD

1990 ◽  
Vol 05 (04) ◽  
pp. 725-746 ◽  
Author(s):  
A. FOUSSATS ◽  
O. ZANDRON
Keyword(s):  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Covariant Formalism ◽  
Field Equations ◽  
Bianchi Identities ◽  
First Order ◽  
Group Manifold

By considering two explicit examples D=6 and D=11 supergravities we show the applicability, and check the validity of the first order canonical covariant formalism on group manifold. In each case we find the primary constraints, the total Hamiltonian and the field equations of motion. Moreover, we show how the Bianchi identities appear in the covariant Hamiltonian formalism.

scholarly journals Exploring Alternatives to the Hamiltonian Calculation of the Ashtekar-Olmedo-Singh Black Hole Solution

2021 ◽  
Vol 8 ◽  
Author(s):  
Alejandro García-Quismondo ◽  
Guillermo A. Mena Marugán
Keyword(s):  
Black Hole ◽  
Phase Space ◽  
Black Hole Solution ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Quantum Geometry ◽  
Dynamical Equations ◽  
Constants Of Motion ◽  
Geometry Effects ◽  
Hole Model

In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit.

Sign in / Sign up

Export Citation Format

Share Document